Momentum
Contents: 8 sections
Momentum
momentum = mass x velocity, so p = mv
Units are kg m/s. There is no special name for this unit, which is a small help: if your answer carries any other unit, the working is wrong.
Momentum is a vector. It inherits its direction from the velocity, and that is the single most important fact in this subtopic. Mass is a scalar; multiplying a vector by a scalar leaves a vector.
The equation is a product, not a quotient. An option reading p = m / v is not a physical quantity at all, and its units, kg divided by m/s, are nothing you will ever meet.
Why the direction matters
Take one direction as positive and hold it for the whole question. Anything moving the other way carries a negative momentum.
This is what allows momentum to cancel. Two objects can approach each other with equal and opposite momenta, collide, and both stop, without any momentum disappearing: the total was zero before and is zero after.
It is also why a bouncing ball is the standard question.
Worked example. A 0.50 kg ball hits the floor at 10 m/s and rebounds at 8.0 m/s. The collision lasts 0.50 s. Find the average force.
Take upwards as positive. Initial velocity = −10 m/s. Final velocity = +8.0 m/s. Change in velocity = 8.0 − (−10) = 18 m/s upwards. Change in momentum = 0.50 x 18 = 9.0 kg m/s upwards. Force = 9.0 / 0.50 = 18 N upwards.
Subtracting the speeds as though both were in the same direction gives 2.0 m/s and an answer nine times too small, and that answer is always on offer. When anything bounces the speeds add, because the object has to be stopped and then driven back the other way.
The force is upwards because the floor has to push the ball back.
Force as rate of change of momentum
resultant force = change in momentum / time taken
This is Newton's second law in its more general form, and F = ma is the special case where the mass is constant.
Rearranged, force x time = change in momentum. The product force x time is the impulse.
impulse = Ft = change in momentum = mv − mu
The practical consequence is the reason for crumple zones, airbags, crash mats and bending your knees when you land. The change in momentum is fixed by the collision, so increasing the time of the collision reduces the force. Nothing reduces the momentum change; you can only spread it out.
Conservation of momentum
In the absence of external forces, the total momentum before a collision equals the total momentum after.
total momentum before = total momentum after
This holds for collisions where the objects bounce apart and for those where they stick together.
Worked example. A 0.16 kg ball moving at 0.50 m/s strikes a stationary 0.10 kg ball. The second ball moves off at 0.50 m/s. Find the speed of the first ball afterwards.
Before: 0.16 x 0.50 + 0.10 x 0 = 0.080 kg m/s. After, the second ball carries 0.10 x 0.50 = 0.050 kg m/s. So the first ball carries 0.080 − 0.050 = 0.030 kg m/s. Its speed = 0.030 / 0.16 = 0.19 m/s.
Work in momentum throughout and convert back to a speed only at the very end.
The first ball stops dead only when the two masses are equal, which is a special case and not the general rule.
Momentum and kinetic energy
Momentum is conserved in every collision. Kinetic energy is not.
- In an elastic collision, kinetic energy is also conserved.
- In an inelastic collision, some kinetic energy becomes thermal energy, sound, or work done deforming the objects. Momentum is still conserved.
Most real collisions are inelastic. If a question mentions that the objects make a noise, or that they stick together, or that they are deformed, it is telling you kinetic energy is not conserved, and any option claiming it is can be struck out.
Newton's third law in this topic
The force each object exerts on the other is equal and opposite, and they act for the same length of time. So the impulses are equal and opposite too, and one object gains exactly the momentum the other loses.
That is why statements such as "the change in momentum of X equals the change in momentum of Y" are wrong as written. The changes are equal in size and opposite in direction, and dropping the minus sign changes the meaning.
Common mistakes
- Subtracting the speeds in a rebound instead of adding them.
- Forgetting that momentum is a vector and ignoring signs.
- Saying kinetic energy is conserved in a collision the question has told you is noisy or sticky.
- Writing the changes in momentum of two colliding objects as equal without the opposite sign.
- Assuming the incoming object always stops after a collision.
- Quoting momentum in newtons, or impulse in kg m/s² .
- Thinking a crumple zone reduces the change in momentum rather than the force.
Check you have it
Question 1
A ball of mass m falls vertically and hits a hard surface.
Its speed on hitting the surface is v1.
It rebounds vertically upwards with speed v2.
What is the change in momentum of the ball?
Answer: C.
Take downwards as positive. The ball arrives at +v1 and leaves at -v2, so:
change in velocity = -v2 - v1, a magnitude of (v1 + v2)
change in momentum = m(v1 + v2), which is C.
D, m(v2 - v1), is the trap and it is the answer you get by subtracting the speeds as though the ball carried on in the same direction. It would be right if the ball passed through the surface and kept going, and it gives a much smaller number, sometimes even a negative one.
A and B each account for only one half of the journey, before or after, and ignore the other.
The physical point is worth holding on to: a ball that bounces changes its momentum by more than one that simply stops dead, because stopping removes mv1 while bouncing removes mv1 and then adds mv2 in the opposite direction. That is why a bouncing ball pushes harder on the floor than a lump of putty of the same mass and speed.
Question 2
An object is moving at +3.0 m / s.
A force acts on the object.
After a time, the object is moving at –4.0 m / s.
The mass of the object is 5.0 kg.
What is the change in momentum of the body?
Answer: A.
change in momentum = m x (v - u) = 5.0 x (-4.0 - 3.0) = 5.0 x (-7.0) = -35 kg m / s, which is A.
D, +35, has the right size and the wrong sign. The change is negative because the momentum ended up pointing in the negative direction, and the question gives signed velocities precisely so the sign matters.
B and C, plus or minus 5.0, come from subtracting 4.0 from 3.0 and ignoring the minus sign on the final velocity. That treats the object as having slowed from 3.0 to 4.0 in the same direction, which is not what happened and is not even a decrease.
Whenever a velocity is written with a minus sign, substitute it complete with its sign and let the arithmetic handle the direction.
Question 3
A resultant force F accelerates a car of mass m along a straight horizontal road from rest to a speed v in time t, giving it momentum p.
Which pair of relationships for this situation is correct?
Answer: C.
p = mv is the definition of momentum: mass times velocity. Correct.
Ft = p is the impulse relationship. Force times time equals the change in momentum, and here the car starts from rest, so the change is the final momentum p. Correct.
Both right, so C.
A has pt = mv, which multiplies momentum by time for no reason, and F = pt, which multiplies rather than divides.
B pairs the correct p = mv with F = pt. That is wrong by units: force is momentum divided by time, not multiplied by it. Momentum is in N s, so dividing by seconds gives newtons, while multiplying gives N s squared.
D has p = mvt, which adds a stray time to the definition of momentum.
Units are the quickest way through algebra questions of this shape. Check each expression's units before checking its physics, and three of these four fall over immediately.
What the syllabus asks for on this topicSyllabus points
Syllabus points
- Define momentum and recall p = mv.
- Define impulse and recall that impulse equals the change in momentum.
- Apply the principle of conservation of momentum to collisions in one dimension.
- Define resultant force as the change in momentum per unit time.
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